# Tagged Questions

**-1**

votes

**0**answers

72 views

### Long cycles in $\text{Sym}(n)$ [on hold]

Let $x,y \in \text{Sym}(n)$ (symmetric group on $\{ 1,2,\ldots,n \}$) and $z:=xy$.
Question: What non-trivial sufficient conditions (on $x$ and $y$) for $z$ to be a cycle of length $n$ do we know?
...

**2**

votes

**1**answer

128 views

### counting the number of ordered pairs in a permutohedron

Recall that a permutohedron is a graph on the set of permutations $S_n$ with an edge between $\sigma$ and $\tau$ if they differ by one adjacent transposition: $\tau = (i,i+1) \circ \sigma$ for some $i ...

**17**

votes

**5**answers

536 views

### Why is the right permutohedron order (aka weak order) on $S_n$ a lattice?

This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the ...

**9**

votes

**3**answers

561 views

### How do most people write permutations?

I'd like to know how people prefer to write permutations, or elements of the symmetric group $S_n$ for $n\ge0$.
The most natural way to define a permutation in $S_n$ is as a bijection on the set ...

**9**

votes

**4**answers

848 views

### Number of Permutations?

Edit: This is a modest rephrasing of the question as originally stated below the fold: for $n \geq 3$, let $\sigma \in S_n$ be a fixed-point-free permutation. How many fixed-point-free permutations ...

**3**

votes

**1**answer

247 views

### Distances on generalizations of the symmetric group

I'm a computer vision student, and I'm looking for some symmetric group literature guidance. I'm going to provide some context, and finally ask two questions.
The Cayley distance and other distances ...

**17**

votes

**4**answers

1k views

### An n!xn! determinant.

Let us consider the matrix $A$ with its rows and columns enumerated by the elements of $S_n$ with $A_{\sigma\tau}=x^{c(\sigma\tau^{-1})}$ where $c()$ is the number of cycles in a permutation's ...

**9**

votes

**4**answers

1k views

### Cyclic Permutations - but not what you think

This question is not about elements of $S_n$ that consist of a single $n$-cycle, though naturally it's related.
Instead, consider permutations modulo the action of $(123\ldots n)$. That is, we ...

**6**

votes

**3**answers

1k views

### Sarrus determinant rule: references, extensions

SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically ...